Mixed domain systems are those in which one signal is analog and the other is digital. In characterizing the response of such systems (i.e. the ratio of the output signal spectrum to the input linear signal spectrum), a dual channel signal analyzer may be used, with the digital signal being applied directly to one of the analyzer input channels, and with the analog signal being applied to a second analyzer input channel through an analog-to-digital converter internal to the instrument.
A twofold difficulty arises is making measurements on mixed domain systems. The first is in synchronizing the analog signal sample points with an external clock associated with the device under test used to sample the digital signal. Such synchronization is required to maintain phase accuracy in the resulting measurements. The second difficulty is in providing alias protection for the sampled analog signal. This latter difficulty relates to the phenomenon inherent in sampled data systems wherein the spectrum of a sampled analog signal below one half the sampling frequency is replicated (mirrored) repeatedly around it.
To eliminate the aliasing problem, one approach is to filter the analog signal prior to sampling with an anti-alias filter so as to insure that all spectral components above half the sampling frequency (which would otherwise be mirrored back into the basedband spectrum) are filtered out. However, this is impractical in a general purpose measurement system since the sampling frequency may range over several decades (perhaps 1 Hz to more than 25 KHz). It is difficult and expensive to design an analog filter that can be adaptable to such a broad range of cutoff frequencies.
FIG. 1 illustrates a representative mixed domain system 10 in which these difficulties are encountered, and illustrates a form of the present invention applied thereto. The analysis instrument 12 is a dual channel FFT instrument, such as the Hewlett-Packard 3563A, in which a microprocessor 14 (with associated program ROM 16 and scratchpad RAM 18) performs Fast Fourier Transforms (FFTs) on two channels of sampled input data. These transforms yield Fourier coefficients which are accumulated in "bins," one of which typically corresponds to each element of resolution on a spectral display 20 associated with the instrument. (FFT instruments are disclosed in greater detail in U.S. Pat. Nos. 4,932,062, 4,928,251, 4,918,381, 4,755,795 and 4,713,782, the disclosures of which are incorporated herein by reference.)
The analysis instrument 12 includes a digital signal source 22 that outputs a digital excitation signal. This excitation signal is applied to input channel 1 of the instrument as the reference signal, and is also provided to an external digital filter 24, which has a transfer function D*(s). (The superscripted-* symbology refers to the starred transform. This transform is detailed, inter alia, in Phillips, et al, Digital Control System Analysis and Design, Prentice-Hall, pp. 69-75, 106-110 (1984). The non-superscripted-* symbology refers to a convolution operation.) The filter 24 operates with a clock frequency of F.sub.s digital, and has a delay equal to t.sub.0 seconds. The filtered digital signal is converted into analog form by a digital-to-analog converter 26, which has a transfer function of e.sup.-sto [(1-e.sup.-sT)/s]. (All digital filters require a finite amount of time to produce a result. This computational delay is modelled here as part of the digital-to-analog converter 26.)
The analog signal from the digital-to-analog converter 26 is applied to an analog circuit 28, which has a transfer function G(s). (The circuit 28 may, for example, be an analog filter circuit.) The analog output signal from circuit 28 is applied to input channel 2 of the instrument, where it is filtered by a low-pass filter 30. The filtered analog signal is converted into digital form for analysis by an analog-to-digital converter (ADC) 32 that samples the analog signal at a frequency F.sub.s analog. (In the illustrated embodiment, all of the clocking/sampling signals are provided by a common clock circuit 34. In other embodiments, the clocking signals used in the mixed domain system 10 need not be synchronized with the analog and digital sampling signals.) The samples applied to the first and second input channels are clocked into a pair of 2048 byte (13-bit bytes) memories 36, 38 internal to the instrument. It is on the sampled data stored in these two memories that the FFT analyses are performed to determine the system transfer function.
The transfer function of the overall FIG. 1 system 10 can be described as follows: EQU Y(s)/E*(s)=G(s)D*(s)e.sup.-sto [(1-e.sup.-sT)/s] (1)
where E*(s) is the spectrum of the digital signal source 22, and Y(s) is the spectrum output by the analog circuit 28. (The term "transfer function" is not a technically accurate descriptor of this function due to its mixed domain character. It may more properly be considered a ratio of linear spectrums having practical significance.)
The analysis instrument 12 operates to divide out the E*(s) signal applied to channel 1 from the resultant signal applied to channel 2. However, this E*(s) signal is modified prior to apllication to the analog circuit 28 by the intervening digital filter 24, with its transfer function of D*(s). D*(s) has a periodic spectrum over 1/T=F.sub.s digital. The resulting spectrum is further modified by the digital-to-analog converter 26, with its transfer function e.sup.-sto [(1-e.sup.-sT)/s].
The transfer function of the digital-to-analog converter 26 has a sin(x)/x rolloff (due to its sample and hold operation) which does not filter the replicated spectrum coming from D*(s) beyond F.sub.s digital /2. Thus, if converter 32 samples at the same rate as converter 26, channel 2 will be an aliased measurement. As noted, it is impractical to design a low pass filter whose cutoff frequency can always be adjusted to F.sub.s digital /2, since this represents a wide range of possible frequencies.
In accordance with one embodiment of the present invention, the foregoing aliasing and synchronization difficulties are overcome by sampling the analog channel at an integer multiple of the digital channel, and zero filling the set of sampled digital data so that it corresponds to the more densely sampled analog data. By so doing, measurements across a mixed domain boundary (such as the illustrated digital-to-analog converter 26) can be made to determine frequency response above the classical F.sub.s digital /2 Nyquist limit.
The foregoing and additional features and advantages of the present invention will be more readily apparent from the following detailed description, which proceeds with reference to the accompanying drawings.